1.9.2: Exercise N.9
- Page ID
- 147916
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) In December 2020, the FDA published a report that included the accuracy of the Ellume COVID-19 Home Test. The article reported that the test, “correctly identified 96 percent of positive samples and 100% of negative samples in individuals with symptoms.”30
(a) What do the percentages in the paragraph tell us about the specificity and sensitivity of the test?
(b) Suppose that this test was administered by a group of 10,000 people, of which 1% are infected with COVID-19. Fill in the following table in order to gather data on the test’s effectiveness.
Hint: How many people in total have COVID-19? Where does that go in the table? How many of those would test positive?
| Number of COVID-19 Positive Individuals | Number of COVID-19 Negative Individuals | Total | |
| Result of Ellume COVID-19 Test is Positive | |||
| Result of Ellume COVID-19 Test is Negative | |||
| Total |
(c) How many individuals would receive a positive test result even though they are not infected with COVID-19? Based on the above table, if someone is not infected with COVID-19, what is the probability that they will receive a positive test result?
(d) How many of the negative test results would be received by people who were infected with COVID-19? Based on the above table, if someone receives a negative result from this new test, what is the probability that the person is actually infected with COVID-19? Round to the nearest hundredth of a percentage point.
(e) Suppose that this same test was administered to 10,000 people in another country in which 10% of the population is infected with COVID-19. Fill in the following table:
| Number of COVID-19 Positive Individuals | Number of COVID-19 Negative Individuals | |
| Result of Ellume COVID-19 Test is Positive | ||
| Result of Ellume COVID-19 Test is Negative |
(f) In this population, if someone tests positive with this new test, what is the probability that the person is not infected with COVID-19? Round to the nearest hundredth of a percentage point.
(g) In this population, if someone tests negative with this new test, what is the probability that the person is infected with COVID-19? Round to the nearest hundredth of a percentage point.
(2) Suppose an even newer test for COVID-19, released in 2023, advertises that it has improved sensitivity compared to the test described in Question 1. The advertisement then states that people should use the newer test because it is more effective. The two-way table below shows the data on which the advertisement’s claims were based.
| Number of COVID-19 Positive Individuals | Number of COVID-19 Negative Individuals | |
| Result of COVID-19 Test is Positive | 49 | 40 |
| Result of COVID-19 Test is Negative | 1 | 19910 |
Calculate the sensitivity and specificity of the newer test. Round to the nearest tenth of a percentage point.
(3) Consider the following situation presented in a New York Times blog: “Before going on vacation for a week, you ask your spacey friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent.”31 We can model this situation by using a hypothetical 100 ailing plants and entering what would happen into a two-way table like the one below:
| Watered | Not Watered | Total | |
| Number of Plants that Died | |||
| Number of Plants that Did Not Die | |||
| Total | 100 |
(a) Fill in the table using the percentages in the scenario. (Hint: Start with column totals: watered and not watered.)
(b) Now you can answer the questions the New York Times blogger asks. Round to the nearest percentage point.
(i) What’s the chance that your plant will survive the week?
(ii) If the plant is dead when you return, what’s the chance that your friend forgot to water it?
(iii) If your friend forgot to water the plant, what’s the chance it’ll be dead when you return?
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31 http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/